Consider the given vector equation. r(t) = (3t – 3, t2 + 4) (a) Find r'(t). r'(t) = (b) Sketch the plane curve together with the position vector r(t) and the tangent vector r'(t) for the given value of t = -2.

Q: Find the position vector for a particle with acceleration, initial velocity, and initial position…

A:

Q: find a vector normal to the plane with the given equation. x − z = 0

A: Given: Plane: x-z=0 To Find: A vector normal to the given plane

Q: Find a vector normal to the plane with the given equation. 3(x – 5) – 5(y – 1) + 10z = 0 n =

A: Consider the equation of the plane passing through a general point Po and with normal vector n.

Q: Sketch the curve of the vector function r(t) = 3ti+ (3t + 1)j,-1<ts2, and indicate the orientation…

A: The given curve in vector form: r(t)=3ti+(3t+1)j The point of t is -1≤t≤2 To find the curve at the…

Q: Find the parametric equations of the tangent line to the curve represented by the vector function…

A:

Q: Suppose you are given a vector-valued function modelling the velocity of a particle in space. At…

A:

Q: What is the vector parametric equation L(t) for the line through the points (5,-5,0) and (0,-3,3)?

A:

Q: Find the vector equation of the plane tangent to the surface z = x^3 - y^3 at the point (0,1,-1)

A:

Q: At time to, a moving particle has the velocity vector v = 7i and the acceleration vector a = 7i +…

A:

Q: Sketch the curve parametrized by r(f) = (1 – 12, 1) for -1sı<1. Compute the tangent vector at i = 1…

A: Let's first draw the curve -  Domain - -1 ≤ t ≤ 1 where, we can take values of t as -  t = -1, -0.5,…

Q: Find the tangential and normal components of acceleration for the position vector denoted by 7 = (t…

A: We have given position vector  t+29,2t,t2 Then, Tangential component of acceleration aT=r't.r''tr't…

Q: give the position vectors of particles moving alongvarious curves in the xy-plane. In each case,…

A:

Q: Represent the plane curve by a vector-valued function. x- + y2 = 4 r(t) = 2 cos ti +2 sin tj

A: x2 + y2 = 4 x2 + y2 = 22 = r2 Therefore, r = 2

Q: What is the directional derivative of the function f(x, y, z) = 2xy + z² at the point (1, –1,3) in…

A: The Directional Derivative: Let the function f(x, y) be the height of a mountain range at each point…

Q: Sketch the curve represented by the vector-valued function                               r(t) = 2…

A: Given information: Vector-valued function rt=2costi+tj+2sintk trigonometric identity, sin2t+cos2t=1…

Q: Sketch the curve with vector equation r(t) = (t2 – 2, t) in the xy-plane. Indicate with arrows the…

A:

Q: 3. A unit vector tangent to the curve y x2 at the point (2, 4).

A: We are entitled to solve only one question at a time so we will answer only the first question.…

Q: Q2. A point is moving along a curve whose position vector is Find : the acceleration components at /…

A: Consider the given position vector

Q: Find the Tangent vector, the Normal vector, and the Binormal vector (T, N and B) for the curve F(t)…

A:

Q: Sketch the path of an object moving along the plane curve given by                  r(t) = (t2 − 4)i…

A: To sketch the path of an object moving along the plane curve given by r(t)=(t2-4)i+tj To find the…

Q: Sketch the curve represented by the vector-valued function                               r(t) = ⟨…

A: Given  r(t) = ⟨ cos t, sin t ⟩

Q: x-1 y-2 z-1 2 -6 7 1 F(t) = ( 21 + ¹ )i + ¹/ (m² − 3)} j + t√21³-1k at t=1 1 4 (c) show that the…

A:

Q: It is known that the acceleration vector for a particle is given by A(t) = 4j + 2tk. Determine the…

A: We need to determine vector equation of curve traced.

Q: Find the equation of the tangent line of the vector function F(t) = (e= cos(t), e=' sin(t), e') at…

A:

Q: Find the position vector for a particle with acceleration, initial velocity, and initial position…

A:

Q: Find the directional derivative of the function f(x, y) = 2x2y + 6xy at (1, 1) in the direction of a…

A:

Q: Find a vector tangent to the curve of intersection of the two cyclinders x² + y? = 2 and y? + z² = 2…

A: To find a vector tangent to the curve of intersection of two cylinders        x2+y2=2 and…

Q: The angular velocity vector, ω, can be obtained by adding all components of the angular velocities…

A: Given, Angular velocity vectorω as the addition of all angular velocities. To find: To choose the…

Q: Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t…

A:

Q: 2. If a particle moves in the xy-plane so that at any time t > 0, its position vector is (In(r² +…

A: Solution-

Q: Find the directional derivative of the function f(x, y,z) =x-y+2z at the point P(1,2,3) in the…

A: Note:- Let a function f(x, y, z), then Direction Derivative of the function f at point P in…

Q: For time >0, the position of a particle moving in the xy-plane is given by the parametric equations…

A:

Q: A bee with velocity vector r' (t) starts out at (2, 5, -2) at t = 0 and flies around for 1 seconds.…

A: We can answer first question as per the honor policy. Please resubmit other parts as well to know…

Q: V3 t², a cost + at sin t) 2 r(t) = (a sint – at cos t,

A:

Q: Find the directional derivative of the function f = In (x² + y2) at the point P(4,5) in the…

A: By Bartleby policy I have to solve only first one as Q)3)a) and Q)3)b) are unrelated  problems.These…

Q: A particle moves through 3-space in such a way that its acceleration is a(t) = 8sin 2ti +8 cos…

A:

Q: The directional derivative of f(x, y) = xe + cosxy at P,(2,0) in the direction of the vector 3i - 4j…

A:

Q: The equation of the tangent plane to this ellipsoid at (1, –2, –2) is The vector parametric form of…

A: Let's find.

Q: Find the parametric equations of the tangent line to the curve represented by the vector function 7…

A: Topic  = Vector

Q: Find a unit vector that is normal to the level curve of the function f(x, y) = 5x + 6y at the point…

A:

Q: Find a vector equation of the line tangent to the graph of r(f) at the point Po on the curve. r(1) =…

A: Find the derivative of the function r(t) to find the equation tangent to the graph of the function.…

Q: Find the position vector for the particle with acceleration, initial velocity, and initial postion…

A:

Q: Find the position vector for a particle with acceleration, initial velocity, and tinitial position…

A:

Q: Find the position vector for a particle with acceleration, initial velocity, and initial position…

A:

Q: find the velocity and acceleration vectors in terms ofur and uθ . r = a(1 + sin t) and θ = 1 - e-t

A: The velocity and acceleration in terms of ur→ and uθ→ are given by: v→=r'ur→+r θ'uθ→a→=r''-r…

Q: (iii) Find a vector equation for the tangent line to the curve of intersection of the cylinder a² +…

A:

Q: 4. If a particle moves in the xy-plane so that at time t its position vector is sin 31 ,3r2 find the…

A:

Q: Let r(t) = (sin(2t), 3t, cos(2t)), t E [-T, "] be the position vector of a particle at time t. (a)…

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: What is the directional derivative of f(x, y) = 4xy+ y cos x at the point (0, 1) in the direction of…

A: